Relations among starlikeness, convexity and \(k\)-fold symmetry of locally biholomorphic mappings in \(\mathbb{C}^n\)
The authors study certain connections between star-like mappings and mappings with \(k\)-fold symmetry in \(\mathbb{C}^n\). A domain \(\Omega\subset\mathbb{C}^n\) is said to be \(k\)-symmetric if \(\epsilon\Omega=\Omega\) for \(\epsilon=e^{\frac{2\pi i}{k}}\). A function \(g:\Omega\rightarrow\mathbb{C}^m\) defined on such a domain is said to be \((j,k)\)-symmetric if \(g(\epsilon z)=\epsilon^j g(z)\) for all \(z\in \Omega\). A generalization to higher dimensions of the classical decomposition of a real-valued function into its even and odd parts, asserts that \[ f=\sum_{j=0}^{k-1} f_{j,k} \, , \] where the \(f_{j,k}\) are \((j,k)\)-symmetric. The paper recalls the well-known characterization of star-likeness in the ball \(\mathbb{B}^n\): \[ f(z)=Df(z)h(z) \, , \] for some holomorphic \(h\) with \[ \text{Re}\langle h(z),z\rangle > 0 \;\, , \; z\in\mathbb{B}^n\backslash\{0\}\, , \eqno (1) \] and discusses to what extent this characterization carries over to the \((1,k)\)-symmetric component of \(f\). More specifically, the authors consider the class \(S(k)\), \(k\geq 2\), of locally biholomorphic mappings \(f:\mathbb{B}^n\rightarrow\mathbb{C}^n\) with \(f(0)=0\), \(Df(0)=I\), for which \[ f_{1,k}(z)=Df(z)h(z) \, , \eqno(2) \] for \(h\) satisfying (1). If \(f\) is star-like and \((1,k)\)-symmetric, then equation (2) will hold with \(h\) which is also \((1,k)\)-symmetric [the author, J. Math. Anal. Appl. 281, No. 1, 276--286 (2003; Zbl 1029.32006)]. Conversely, if equation (2) holds for a holomorphic \((1,k)\)-symmetric \(h\) that satisfies (1), then \(f\) is star-like and \((1,k)\)-symmetric [loc. cit.]. The authors then proceed to study various inclusions between the families of star-like and convex mappings, and the class \(S(k)\). Finally they investigate global properties of the class \(S(k)\), such as convexity and connectedness.
scientific article; zbMATH DE number 6684632
scientific article